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The role of mixed-layer instabilities in submesoscale turbulence

Published online by Cambridge University Press:  22 December 2015

Jörn Callies*
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Glenn Flierl
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Raffaele Ferrari
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Baylor Fox-Kemper
Department of Earth, Environmental and Planetary Sciences, Brown University, 324 Brook Street, Providence, RI 02912, USA
Email address for correspondence:


Upper-ocean turbulence at scales smaller than the mesoscale is believed to exchange surface and thermocline waters, which plays an important role in both physical and biogeochemical budgets. But what energizes this submesoscale turbulence remains a topic of debate. Two mechanisms have been proposed: mesoscale-driven surface frontogenesis and baroclinic mixed-layer instabilities. The goal here is to understand the differences between the dynamics of these two mechanisms, using a simple quasi-geostrophic model. The essence of mesoscale-driven surface frontogenesis is captured by the well-known surface quasi-geostrophic model, which describes the sharpening of surface buoyancy gradients and the subsequent breakup in secondary roll-up instabilities. We formulate a similarly archetypical Eady-like model of submesoscale turbulence induced by mixed-layer instabilities. The model captures the scale and structure of this baroclinic instability in the mixed layer. A wide range of scales are energized through a turbulent inverse cascade of kinetic energy that is fuelled by the submesoscale mixed-layer instability. Major differences to mesoscale-driven surface frontogenesis are that mixed-layer instabilities energize the entire depth of the mixed layer and produce larger vertical velocities. The distribution of energy across scales and in the vertical produced by our simple model of mixed-layer instabilities compares favourably to observations of energetic wintertime submesoscale flows, suggesting that it captures the leading-order balanced dynamics of these flows. The dynamics described here in an oceanographic context have potential applications to other geophysical fluids with layers of different stratifications.

© 2016 Cambridge University Press 

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Badin, G. 2012 Surface semi-geostrophic dynamics in the ocean. Geophys. Astrophys. Fluid Dyn. 107 (5), 526540.CrossRefGoogle Scholar
Bishop, C. H. 1993a On the behaviour of baroclinic waves undergoing horizontal deformation. I: The ‘RT’ phase diagram. Q. J. R. Meteorol. Soc. 119 (510), 221240.Google Scholar
Bishop, C. H. 1993b On the behaviour of baroclinic waves undergoing horizontal deformation. II: Error-bound amplification and Rossby wave diagnostics. Q. J. R. Meteorol. Soc. 119 (510), 241267.Google Scholar
Blumen, W. 1978 Uniform potential vorticity flow: Part I. Theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35 (5), 774783.2.0.CO;2>CrossRefGoogle Scholar
Blumen, W. 1979 On short-wave baroclinic instability. J. Atmos. Sci. 36 (10), 19251933.2.0.CO;2>CrossRefGoogle Scholar
Boccaletti, G., Ferrari, R. & Fox-Kemper, B. 2007 Mixed layer instabilities and restratification. J. Phys. Oceanogr. 37 (9), 22282250.CrossRefGoogle Scholar
Boyd, J. P. 1992 The energy spectrum of fronts: time evolution of shocks in Burgers’ equation. J. Atmos. Sci. 49 (2), 128139.2.0.CO;2>CrossRefGoogle Scholar
Bretherton, F. P. 1966 Critical layer instability in baroclinic flows. Q. J. R. Meteorol. Soc. 92 (393), 325334.CrossRefGoogle Scholar
Bühler, O., Callies, J. & Ferrari, R. 2014 Wave–vortex decomposition of one-dimensional ship-track data. J. Fluid Mech. 756, 10071026.CrossRefGoogle Scholar
Callies, J. & Ferrari, R. 2013 Interpreting energy and tracer spectra of upper-ocean turbulence in the submesoscale range (1–200 km). J. Phys. Oceanogr. 43 (11), 24562474.CrossRefGoogle Scholar
Callies, J., Ferrari, R., Klymak, J. M. & Gula, J. 2015 Seasonality in submesoscale turbulence. Nat. Commun. 6, 6862.CrossRefGoogle ScholarPubMed
Capet, X., Klein, P., Hua, B. L., Lapeyre, G. & McWilliams, J. C. 2008a Surface kinetic energy transfer in surface quasi-geostrophic flows. J. Fluid Mech. 604, 165174.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008b Mesoscale to submesoscale transition in the California current system. Part I: flow structure, eddy flux, and observational tests. J. Phys. Oceanogr. 38 (1), 2943.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008c Mesoscale to submesoscale transition in the California current system. Part II: frontal processes. J. Phys. Oceanogr. 38 (1), 4464.CrossRefGoogle Scholar
Capet, X., McWilliams, J. C., Molemaker, M. J. & Shchepetkin, A. F. 2008d Mesoscale to submesoscale transition in the California current system. Part III: energy balance and flux. J. Phys. Oceanogr. 38 (10), 22562269.CrossRefGoogle Scholar
Charney, J. G. 1947 The dynamics of long waves in a baroclinic westerly current. J. Meteorol. 4 (5), 135162.2.0.CO;2>CrossRefGoogle Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28 (6), 10871095.2.0.CO;2>CrossRefGoogle Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1 (3), 3352.CrossRefGoogle Scholar
Emanuel, K. A. 1994 Atmospheric Convection. Oxford University Press.Google Scholar
Ferrari, R. 2011 A frontal challenge for climate models. Science 332 (6027), 316317.CrossRefGoogle ScholarPubMed
Ferrari, R. & Rudnick, D. L. 2000 Thermohaline variability in the upper ocean. J. Geophys. Res. 105 (C7), 1685716883.CrossRefGoogle Scholar
Fox-Kemper, B., Ferrari, R. & Hallberg, R. W. 2008 Parameterization of mixed layer eddies. Part I: theory and diagnosis. J. Phys. Oceanogr. 38 (6), 11451165.CrossRefGoogle Scholar
Garner, S. T., Nakamura, N. & Held, I. M. 1992 Nonlinear equilibration of two-dimensional Eady waves: a new perspective. J. Atmos. Sci. 49 (21), 19841996.2.0.CO;2>CrossRefGoogle Scholar
Gill, A. E., Green, J. S. A. & Simmons, A. J. 1974 Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Res. 21 (7), 499528.Google Scholar
Gula, J., Molemaker, M. J. & McWilliams, J. C. 2015 Gulf stream dynamics along the southeastern U.S. seaboard. J. Phys. Oceanogr. 45 (3), 690715.CrossRefGoogle Scholar
Haine, T. W. N. & Marshall, J. 1998 Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. J. Phys. Oceanogr. 28 (4), 634658.2.0.CO;2>CrossRefGoogle Scholar
Hakim, G. J., Snyder, C. & Muraki, D. J. 2002 A new surface model for cyclone–anticyclone asymmetry. J. Atmos. Sci. 59 (16), 24052420.2.0.CO;2>CrossRefGoogle Scholar
Hamlington, P. E., Van Roekel, L. P., Fox-Kemper, B., Julien, K. & Chini, G. P. 2014 Langmuir–submesoscale interactions: descriptive analysis of multiscale frontal spindown simulations. J. Phys. Oceanogr. 44 (9), 22492272.CrossRefGoogle Scholar
Held, I. M., Pierrehumbert, R. T., Garner, S. T. & Swanson, K. L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 120.CrossRefGoogle Scholar
Hoskins, B. J. & Bretherton, F. P. 1972 Atmospheric frontogenesis models: mathematical formulation and solution. J. Atmos. Sci. 29 (1), 1137.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J., Draghici, I. & Davies, H. C. 1978 A new look at the ${\it\omega}$ -equation. Q. J. R. Meteorol. Soc. 104 (439), 3138.CrossRefGoogle Scholar
Juckes, M. 1994 Quasigeostrophic dynamics of the tropopause. J. Atmos. Sci. 51 (19), 27562768.2.0.CO;2>CrossRefGoogle Scholar
Klein, P., Hua, B. L., Lapeyre, G., Capet, X., Le Gentil, S. & Sasaki, H. 2008 Upper ocean turbulence from high-resolution 3D simulations. J. Phys. Oceanogr. 38 (8), 17481763.CrossRefGoogle Scholar
Klein, P. & Lapeyre, G. 2009 The oceanic vertical pump induced by mesoscale and submesoscale turbulence. Annu. Rev. Mater. Sci. 1, 351375.CrossRefGoogle ScholarPubMed
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 301305; (in Russian).Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (7), 14171423.CrossRefGoogle Scholar
Lapeyre, G. & Klein, P. 2006 Dynamics of the upper oceanic layers in terms of surface quasigeostrophy theory. J. Phys. Oceanogr. 36 (2), 165176.CrossRefGoogle Scholar
Lapeyre, G., Klein, P. & Hua, B. L. 2006 Oceanic restratification forced by surface frontogenesis. J. Phys. Oceanogr. 36 (8), 15771590.CrossRefGoogle Scholar
Larichev, V. D. & Held, I. M. 1995 Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr. 25 (10), 22852297.2.0.CO;2>CrossRefGoogle Scholar
Le Traon, P.-Y., Klein, P., Hua, B. L. & Dibarboure, G. 2008 Do altimeter wavenumber spectra agree with the interior or surface quasigeostrophic theory? J. Phys. Oceanogr. 38 (5), 11371142.CrossRefGoogle Scholar
Lévy, M., Iovino, D., Resplandy, L., Klein, P., Madec, G., Tréguier, A.-M., Masson, S. & Takahashi, K. 2012 Large-scale impacts of submesoscale dynamics on phytoplankton: local and remote effects. Ocean Model. 43–44, 7793.CrossRefGoogle Scholar
Lindzen, R. S. 1994 The Eady problem for a basic state with zero PV gradient but ${\it\beta}\neq 0$ . J. Atmos. Sci. 51 (22), 32213226.2.0.CO;2>CrossRefGoogle Scholar
Mahadevan, A. 2014 Eddy effects on biogeochemistry. Nature 506, 168169.CrossRefGoogle ScholarPubMed
Mahadevan, A. & Tandon, A. 2006 An analysis of mechanisms for submesoscale vertical motion at ocean fronts. Ocean Model. 14 (3–4), 241256.CrossRefGoogle Scholar
McWilliams, J. C. & Fox-Kemper, B. 2013 Oceanic wave-balanced surface fronts and filaments. J. Fluid Mech. 730, 464490.CrossRefGoogle Scholar
McWilliams, J. C., Molemaker, M. J. & Olafsdottir, E. I. 2009 Linear fluctuation growth during frontogenesis. J. Phys. Oceanogr. 39 (12), 31113129.CrossRefGoogle Scholar
Mensa, J. A., Garraffo, Z., Griffa, A., Özgökmen, T. M., Haza, A. & Veneziani, M. 2013 Seasonality of the submesoscale dynamics in the Gulf stream region. Ocean Dyn. 63 (8), 923941.CrossRefGoogle Scholar
Molemaker, M. J., McWilliams, J. C. & Capet, X. 2010 Balanced and unbalanced routes to dissipation in an equilibrated Eady flow. J. Fluid Mech. 654 (2010), 3563.CrossRefGoogle Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.CrossRefGoogle Scholar
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6 (3), 273286.CrossRefGoogle Scholar
Pierrehumbert, R. T., Held, I. M. & Swanson, K. L. 1994 Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4 (6), 11111116.CrossRefGoogle Scholar
Qiu, B. 1999 Seasonal eddy field modulation of the north pacific subtropical countercurrent: TOPEX/Poseidon observations and theory. J. Phys. Oceanogr. 29 (10), 24712486.2.0.CO;2>CrossRefGoogle Scholar
Qiu, B. & Chen, S. 2004 Seasonal modulations in the eddy field of the south pacific ocean. J. Phys. Oceanogr. 34 (7), 15151527.2.0.CO;2>CrossRefGoogle Scholar
Rhines, P. B. 1977 The dynamics of unsteady currents. In Sea (ed. Goldberg, E.), vol. VI, pp. 189318. Wiley.Google Scholar
Richman, J. G., Arbic, B. K., Shriver, J. F., Metzger, E. J. & Wallcraft, A. J. 2012 Inferring dynamics from the wavenumber spectra of an eddying global ocean model with embedded tides. J. Geophys. Res. 117 (C12), C12012.CrossRefGoogle Scholar
Rivest, C., Davis, C. A. & Farrell, B. F. 1992 Upper-tropospheric synoptic-scale waves. Part I: maintenance as Eady normal modes. J. Atmos. Sci. 49 (22), 21082119.2.0.CO;2>CrossRefGoogle Scholar
Rocha, C. B., Chereskin, T. K., Gille, S. T. & Menemenlis, D.2015 Mesoscale to submesoscale wavenumber spectra in Drake passage. J. Phys. Oceanogr. (in press).Google Scholar
Roullet, G., McWilliams, J. C., Capet, X. & Molemaker, M. J. 2012 Properties of steady geostrophic turbulence with isopycnal outcropping. J. Phys. Oceanogr. 42 (1), 1838.CrossRefGoogle Scholar
Salmon, R. 1978 Two-layer quasi-geostrophic turbulence in a simple special case. Geophys. Astrophys. Fluid Dyn. 10 (1), 2552.CrossRefGoogle Scholar
Sasaki, H., Klein, P., Qiu, B. & Sasai, Y. 2014 Impact of oceanic-scale interactions on the seasonal modulation of ocean dynamics by the atmosphere. Nat. Commun. 5, 5636.CrossRefGoogle Scholar
Scott, R. K. 2006 Local and nonlocal advection of a passive scalar. Phys. Fluids 18 (11).CrossRefGoogle Scholar
Seiff, A., Kirk, D. B., Knight, T. C. D., Young, R. E., Mihalov, J. D., Young, L. A., Milos, F. S., Schubert, G., Blanchard, R. C. & Atkinson, D. 1998 Thermal structure of Jupiter’s atmosphere near the edge of a $5~{\rm\mu}\text{m}$ hot spot in the north equatorial belt. J. Geophys. Res. Planets 103 (103), E10.Google Scholar
Shcherbina, A. Y., D’Asaro, E. A., Lee, C. M., Klymak, J. M., Molemaker, M. J. & McWilliams, J. C. 2013 Statistics of vertical vorticity, divergence, and strain in a developed submesoscale turbulence field. Geophys. Res. Lett. 40 (17), 47064711.CrossRefGoogle Scholar
Smith, K. S. & Bernard, E. 2013 Geostrophic turbulence near rapid changes in stratification. Phys. Fluids 25 (4), 046601.CrossRefGoogle Scholar
Smith, K. S. & Vallis, G. K. 2001 The scales and equilibration of midocean eddies: freely evolving flow. J. Phys. Oceanogr. 31 (2), 554571.2.0.CO;2>CrossRefGoogle Scholar
Smith, K. S. & Vallis, G. K. 2002 The scales and equilibration of midocean eddies: forced–dissipative flow. J. Phys. Oceanogr. 32 (6), 16991720.2.0.CO;2>CrossRefGoogle Scholar
Spall, M. A. 1997 Baroclinic jets in confluent flow. J. Phys. Oceanogr. 27 (6), 10541071.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. H. 1966a Frontogenesis by horizontal wind deformation fields. J. Atmos. Sci. 23 (5), 455465.2.0.CO;2>CrossRefGoogle Scholar
Stone, P. H. 1966b On non-geostrophic baroclinic stability. J. Atmos. Sci. 23 (4), 390400.2.0.CO;2>CrossRefGoogle Scholar
Taylor, J. R. & Ferrari, R. 2010 Buoyancy and wind-driven convection at mixed layer density fronts. J. Phys. Oceanogr. 40 (6), 12221242.CrossRefGoogle Scholar
Thomas, L. N. & Lee, C. M. 2005 Intensification of ocean fronts by down-front winds. J. Phys. Oceanogr. 35 (6), 10861102.CrossRefGoogle Scholar
Thomas, L. N., Tandon, A. & Mahadevan, A. 2008 Submesoscale processes and dynamics. In Ocean Modeling in an Eddying Regime, pp. 1738. American Geophysical Union.CrossRefGoogle Scholar
Thomas, L. N., Taylor, J. R., Ferrari, R. & Joyce, T. M. 2013 Symmetric instability in the Gulf stream. Deep-Sea Res. II 91, 96110.CrossRefGoogle Scholar
Tulloch, R., Marshall, J., Hill, C. & Smith, K. S. 2011 Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean. J. Phys. Oceanogr. 41 (6), 10571076.CrossRefGoogle Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Whitt, D. B. & Thomas, L. N. 2015 Resonant generation and energetics of wind-forced near-inertial motions in a geostrophic flow. J. Phys. Oceanogr. 45 (1), 181208.CrossRefGoogle Scholar
Xie, J.-H. & Vanneste, J. 2015 A generalised-Lagrangian-mean model of the interactions between near-inertial waves and mean flow. J. Fluid Mech. 774, 143169.CrossRefGoogle Scholar
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